Where the paper stops

Kawasaki’s condition is that the alternating sums of the sectors around a vertex each come to a straight angle. Maekawa’s is that the mountains and valleys around it differ by two. The big-little-big lemma is about a sector and both of its neighbours.

Every one of them is a statement about going once around a point. At the edge of the sheet there is nowhere to go around, and the theorems have nothing to say.

What a boundary vertex is not subject to #

A vertex strictly inside the sheet has a full turn of paper around it, divided into sectors by its creases. All four conditions this site checks are about that turn.

A vertex on the sheet’s edge has less than a full turn — often much less, at a corner — and the sectors do not close. Developability, which asks whether the angles sum to a full turn, is not merely satisfied there; it does not apply. Kawasaki’s alternating sum needs a cyclic sequence and there is not one. Maekawa’s difference of two comes from a closed cross-section, and the cross-section at an edge is not closed.

So the conditions are not weaker at the boundary. They are absent. A crease can leave the sheet at any angle it likes, in any assignment, and no local theorem objects.

This is the reason a pattern’s difficulty scales with its interior rather than with its size, and the reason the count worth watching is the number of interior vertices rather than the number of creases.

There is a physical reading of the same statement, and it is the one that makes it feel less like a technicality. A theorem about a vertex is a theorem about paper closing on itself all the way round. Kawasaki says the sectors have to fold onto a line and come back; Maekawa says the cross-section has to close. Both are statements that something goes all the way around and returns. At the edge of the sheet, nothing goes all the way around: the paper simply ends, the sectors fold onto a line and stop, and there is no return trip to be consistent with.

Put that way, the boundary is not an exception to the theorems. It is the place where their hypothesis is unavailable, which is a stronger and less negotiable thing.

Cutting a patch removes equations #

The consequence runs the wrong way round from intuition. Taking a smaller sample of a pattern feels like taking a fair sample. What it actually does is convert interior vertices into boundary vertices, and every conversion deletes four conditions.

A one-cell patch of a tessellation may have no interior vertex at all. A two-by-two patch of the waterbomb has one. A three-by-three has four. The conditions arrive with the interior, at a rate that approaches a constant per cell, and a pattern is only being tested properly once that rate has settled.

The corrugation in the first figure is the cleanest witness this site has. It is a leaf-shaped corrugation tapered the wrong way — narrowing across the fold lines rather than along them — and it was built specifically because it has to fail. It does fail, at eighteen interior vertices, on Kawasaki, with the alternating sums at 186.4° and 173.6°.

As a single row it passes everything. Nothing about the taper changed; the vertices that objected are now on the edge of the paper.

What the corrugation is really saying #

It is worth being exact about why the tapered corrugation fails, because the reason is what makes the one-row patch a fair trap rather than a contrived one.

At an interior vertex of a Miura-like corrugation the two row creases are collinear, so the sector angles depend only on the directions of the column creases above and below — and those are set by the ratio of the zigzag’s shift to the row’s height. Kawasaki’s sums come to a straight angle exactly when the row above and the row below have the same height. The column widths never enter the calculation at all.

So a corrugation may be tapered along its folds and may not be tapered across them, and the difference is invisible in a drawing. Anybody sketching a leaf that narrows toward its tip reaches for the row heights, which is the forbidden one.

Now count the vertices. A one-row corrugation has no interior vertex, so the collinearity that produces the condition has nowhere to be evaluated. Two rows produce the first interior vertices and the failure appears. The taper is wrong in all three drawings and only two of them can tell.

How much of a patch is actually being tested #

“Approaches a constant per cell from below” is the right shape and it is worth having the number, because the number is discouraging.

An $n \times n$ patch of a grid has $(n+1)^2$ vertices of which $(n-1)^2$ are interior, so the share carrying any condition at all is

$$\left(\frac{n-1}{n+1}\right)^2.$$

At two cells across that is 11%. At three, 25%. At four, 36%. Not until $n = 3 + 2\sqrt{2} = 5.83$ — a six-by-six patch — does half of a patch’s vertices come under any theorem in this subject, and reaching 90% requires thirty-eight cells across.

Which is a statement about every patch anybody draws #

That is the finding in its most uncomfortable form. A tessellation patch drawn at a size somebody would print, fold or publish — four cells, six, at most twelve — is a pattern the majority or a large minority of whose vertices are not constrained by anything.

The checker is not being lenient at those vertices. There is no condition there to be lenient about, and the pass it reports is the pass of a test that was never administered.

So the two failures in this essay are the same failure at two sizes. The one-row corrugation hides its taper because it has no interior vertices; the two-by-two waterbomb hides twenty-four rules because it has one interior corner out of four kinds. Both are the boundary’s share being large, and the share is large at every drawable size.

The practical form of that is a ratio to check rather than a rule to remember. Before believing a verdict on a patch, ask what fraction of its vertices carried a condition — and if the answer is a third, the verdict is about a third of the pattern.

The tessellation that hides its own failures #

The same effect, counted rather than illustrated.

The waterbomb tessellation has a natural family of candidate assignments: a letter for the horizontal creases depending on the row’s parity, and a letter for each of the four half-diagonals of a cell depending on the cell’s parity. That is nine bits, so 512 candidate rules, and each can be built and checked exhaustively.

Fifty-six of them pass on a two-by-two patch. Thirty-two pass on three-by-three, four-by-four and five-by-five — the count stops falling, and the survivors are a subset of the earlier set rather than a different set.

The twenty-four that die are the point. They were never foldable. A two-by-two patch of this pattern contains exactly one grid-corner vertex, and those rules fail at a grid-corner vertex the small patch does not have. The conditions did not get stricter; the patch stopped hiding them.

That is a more useful statement than it looks, because it is a test a designer can run. If the number of rules surviving is still falling as the patch grows, the patch is too small. When it stops falling, and the survivors remain a subset, there is reason to believe the count is the pattern’s rather than the sample’s.

Which theorem was checked, and how #

The pattern library asserts both halves of the finding rather than either one.

The corrugation’s refusal is asserted on Kawasaki specifically. A refusal for the wrong reason would be a different bug wearing this one’s clothes, so the check does not merely require that the four-row version fails; it requires that the alternating sums are what fail, and that the one-row version passes.

The rule census is asserted as a shape rather than as a number. The small patch must admit strictly more rules than the large one, the large counts must agree with each other, and the survivors must be a subset. Then one of the rules that died is taken individually, shown passing on the small patch, shown failing on the large one, and the two patches’ interior vertex counts are compared to establish that the difference is where the failure went.

The measurement that ties it together is the ratio in the second figure. It rises, it approaches a ceiling, and the generator refuses to draw if it does not rise — because a sweep in which the constraint density did not change would mean the patch size was irrelevant, which is the opposite of the claim.

The patterns that had shipped before were the wrong shape to find it.

The false failure this site produced #

There is a way to get the boundary wrong that is much worse than ignoring it, and this site made it.

The original test for whether a vertex is interior asked whether any boundary edge was incident on it. That is the same question as “is this vertex on the edge of the sheet” only when the sheet’s edge is subdivided at every point a crease reaches it. It usually is not: a sheet is four long edges, so a crease arriving at the middle of one produces a vertex that lies on the boundary with no boundary edge attached.

Such a vertex was then analysed as though the paper continued past the edge of the paper. Its sectors were read as a partial turn, its alternating sum came out wrong, and the pattern was rejected.

The symptom is a false failure, which is why it survived as long as it did. A gate that wrongly accepts something is discovered when a reader folds it; a gate that wrongly rejects something is discovered when somebody draws a pattern it happens to hit.

The Yoshimura pattern is the case that found it, and it found it because every course runs edge to edge. The test is geometric now: a vertex is on the boundary when it lies on any boundary edge, endpoint or not. Nothing that had shipped before was wrong, because the earlier patterns either subdivide their boundary or reach the edge with a single crease, and a vertex of degree one or two is excluded anyway.

The lesson is about the shape of the mistake rather than the mistake. A boundary test that is slightly too generous rejects valid patterns, and a boundary test that is slightly too mean accepts invalid ones, and only the second is the kind of failure the site’s gates are designed to catch.

The extreme case of a sheet that is mostly edge is worth naming, because it is a whole tractable subject.

What does not get easier #

None of this makes a finite sheet an easier object than an infinite pattern. It makes it a less constrained one, and less constrained is not the same as more foldable.

The conditions removed are local ones, and the local conditions were never the hard part. Whether a whole sheet folds flat is NP-hard and stays NP-hard on a finite patch; the boundary does not help with layer ordering, which is where the difficulty lives, and a patch with a free edge has more freedom in its layer ordering rather than less.

There is a second place the reasoning must not be pushed. A patch that folds says nothing about the pattern, which is this essay’s whole point — but a patch that fails does settle the pattern, immediately and completely. Conditions are necessary, so a pattern containing a vertex that fails contains a failure wherever else it appears. The asymmetry is worth stating plainly: failure on a patch generalises and success does not.

Growing the patch fixes one problem and leaves the other untouched. It supplies the interior the local conditions need, so the conditions become evaluable; it supplies nothing whatever to the question of which layer goes where, and that question is the one that was hard before the patch was cut and is still hard after it has been enlarged.

The infinite pattern, and what a drawing of it is #

A tessellation is properly an infinite object and every drawing of one is a patch. So the question “does this tessellation fold” is never quite the question a figure answers.

What can be established on a finite patch is that the pattern’s repeating unit satisfies the conditions at every vertex type it produces, and that every vertex type appears. The waterbomb has two vertex types — six creases at a grid corner, four at a cell centre — and a three-by-three patch contains both, several times over, in every neighbourhood arrangement the pattern produces. That is the sense in which the census above is about the tessellation rather than about the drawing.

It is also the sense in which it stops. Beyond the local conditions there is the global problem, and the global problem for an infinite pattern is a different question again, on which a patch of any size is silent.

The patch at the centre of the rule census is small enough to draw whole.

Who noticed it, and when #

The distinction between interior and boundary vertices is in the statement of the theorems themselves — Kawasaki’s and Maekawa’s are always stated for interior vertices, and the qualifier is not decoration. What is less often said out loud is the corollary: that a bounded patch of a pattern is a weaker test than the pattern.

It surfaces most sharply in the study of tessellations, where the object of interest is periodic and every picture is finite. The habit of checking a patch large enough to contain every vertex neighbourhood, rather than a patch large enough to look convincing, is the practical form of it.

The engineering form is older and comes from the other direction. Anyone who has built a deployable knows that the edge of a panel array behaves differently from the middle, and that edge effects are where prototypes fail. It is the same fact: the boundary is where the constraints stop, and a structure held together by constraints is loosest where they run out.

The traditional repertoire says it a third way. A great many classical models are built on the preliminary base or its relatives, which have a single interior vertex and a great deal of boundary. That is not an accident of taste. A pattern with one interior vertex has four conditions to satisfy and enormous freedom everywhere else, which is exactly the sort of pattern a person can invent by folding paper rather than by solving anything — and it is why the traditional bases are bases: they are the patterns loose enough to have been found by hand.

The tessellations are the opposite kind of object and arrived much later for that reason. Every added cell adds interior vertices and the freedom collapses, so a tessellation cannot be discovered by trying things. It has to be constructed from a unit that was designed to satisfy the conditions, and then tiled — which is a nineteenth- and twentieth-century way of working rather than a traditional one.

One more place the same reasoning applies #

The argument generalises past patches and past tessellations, and the generalisation is worth a paragraph because it changes what a figure on this site is evidence for.

Any drawing of a pattern is finite. A figure showing four cells of a corrugation is showing a patch, whatever the caption says, and the conditions it demonstrates are the conditions that patch happens to contain. This site’s figures are drawn at sizes chosen to be legible, and legible is a much smaller number than representative.

So there is a standing obligation on a generator here: a figure that argues a pattern folds must be built on a patch large enough to contain every kind of interior vertex the pattern makes, even if the figure then draws a smaller one for the reader. Those are two different sizes and conflating them is exactly the mistake this essay is about. Where a figure shows a small patch, the check behind it was run on a larger one.

Where the ladder goes next #

The immediate continuation is the tessellation the census was run on. The waterbomb tiled is a pattern with two vertex types and an assignment that has to be searched for, and the rule count is its own subject: a unit that folds alone is not a tessellation that folds, and the shortfall is countable.

The other direction is back toward the conditions themselves. If the boundary removes equations, and there were never many patterns that satisfied them anyway, then a small patch is foldable for reasons that have nothing to do with the pattern it was cut from — which is worth remembering the next time a drawing folds on the first try.